Riemannian Manifolds with Integrable Geodesic Flows

In this paper we will survey some recent results on the Hamiltonian dynamics of the geodesic flow of a Riemannian manifold. More specifically, we are interested in those manifolds which admit a Riemannian metric for which the geodesic flow is integrable. In Section 2, we introduce the necessary topics from symplectic geometry and Hamiltonian dynamics (and, in particular, we defined the terms geodesic flow and integrable). In Section 3, we discuss several examples of manifolds which admit metrics whose geodesic flows are integrable, and in Section 4 we consider the topology of manifolds with integrable geodesic flows.